1. Tessellations of the Plane
A tessellation of a plane is the filling of the plane with repetitions of figures in such a way that no figures overlap and there are no gaps.

2. Regular Tessellations
Tessellations with regular polygons are both appealing and interesting because of their simplicity.

3. Regular Tessellations
Which regular polygons can tessellate the plane?

4. Semiregular Tessellations
When more than one type of regular polygon is used and the arrangement of the polygons at each vertex is the same, the tessellation is semiregular.

5. Tessellating with Other Shapes
Although there are only three regular tessellations (square, equilateral triangle, and regular hexagon), many tessellations are not regular. One factor that must always be true is that the sum of the measures of the interior angles around a vertex must equal 360°.

6. Tessellating with Other Shapes
A regular pentagon does not tessellate the plane. However, some non-regular pentagons do.

7. Tessellating with Other Shapes
The following two tessellations were discovered by Marjorie Rice, and the problem of how many types of pentagons tessellate remains unsolved today.

8. Creating Tessellations with Translations
Consider any polygon known to tessellate a plane, such as rectangle ABCD (a). On the left side of the figure draw any shape in the interior of the rectangle (b). Cut this shape from the rectangle and slide it to the right by the slide that takes A to B (c). The resulting shape will tessellate the plane.

9. Creating Tessellations with Rotations
A second method of forming a tessellation involves a series of rotations of parts of a figure. Start with an equilateral triangle ABC (a), choose the midpoint O of one side of the triangle, and cut out a shape (b), being careful not to cut away more than half of angle B, and then rotate the shape clockwise around point O.

10. Who is M.C. Eshcer? Born 1898 Always Sick so he drew all the time.
He Loved Math and Art. So he made drawings using mathematical formulas.

11. M.C. Escher created many tessellations throughout his life.

12. Today we are making Tessellations
Rule #1: The tessellation must “tile a floor” that goes on forever, with no overlapping and no gaps.

13. STEP #1: You will get a square

14. STEP #2: You will draw a shape from one top corner to the other top corner.

15. STEP #3: Move the new shape to the bottom and tape it.

16. STEP #4: Draw a second shape from one side corner to the other side corner.

17. STEP #5: Move and tape the new shape to the opposite side.

18. Now your shape should look something like this.

19. STEP #6: Start to “tile your plane”